An algorithm producing hyperbolicity equations for a link complement in \(S^ 3\)
type: Item
instance of: scholarly article
title: An algorithm producing hyperbolicity equations for a link complement in \(S^ 3\)
zbMATH Open document ID: 0769.57011
DOI: 10.1007/BF00147745
author: Carlo Petronio
published in: Geometriae Dedicata
publication date: 1993-04-01T00:00:00Z
review text: An orientable non-compact hyperbolic 3-manifold of finite volume can be decomposed into a finite number of ideal tetrahedra with faces glued together in pairs. These tetrahedra are parametrized by complex numbers satisfying certain rational equations depending on the combinatorics of the gluing. The author produces an algorithm for writing down these hyperbolicity equations for a class of link complements in the 3-sphere. His algorithm refines one described in work of W. Menasco. The class of links to which the algorithm applies are those with a planar projection to \(R^ 2\) which satisfies the following conditions: it is alternating, any two edges with the same endpoints bound a region, its complementary regions are discs and the unbounded region has at least 3 vertices, and it is not the obvious projection of an \((n,2)\) torus knot or link or the unknotted circle. The author shows that by adding additional circles, any link can be enlarged to one to which the algorithm applies. The algorithm is carefully explained with 54 accompanying figures.
reviewed by: Darryl McCullough
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 147288
zbMATH Keywords: orientable non-compact hyperbolic 3-manifold of finite volume
zbMATH Keywords: ideal tetrahedra
zbMATH Keywords: gluing
zbMATH Keywords: hyperbolicity equations
zbMATH Keywords: link complements
MaRDI profile type: Publication
Recommended article: Ideal triangulations of link complements and hyperbolicity equations
Recommended article: An alternative approach to hyperbolic structures on link complements
Recommended article: A polyhedral approach to the arithmetic and geometry of hyperbolic link complements
Recommended article: Algorithmic construction and recognition of hyperbolic 3-manifolds, links, and graphs
Recommended article: On identifying hyperbolic 3-manifolds as link complements in the 3-sphere
Recommended article: An effective algorithm to get linking numbers of 3-manifolds
Recommended article: A geometrically bounding hyperbolic link complement
Recommended article: Q2973958
Recommended article: Q4320575
Recommended article: Q3378443
description: scientific article; zbMATH DE number 147288